User anton fonarev - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T09:34:01Z http://mathoverflow.net/feeds/user/10941 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/132965/normal-bundle-of-an-exceptional-divisor/133046#133046 Answer by Anton Fonarev for Normal bundle of an exceptional divisor Anton Fonarev 2013-06-07T08:38:13Z 2013-06-07T08:38:13Z <p>As for the first question, this is written in many places. Searching MO, by the way, is a good idea: <a href="http://mathoverflow.net/questions/16788/blowing-up-1-curves-effective-and-ample-divisors" rel="nofollow">http://mathoverflow.net/questions/16788/blowing-up-1-curves-effective-and-ample-divisors</a></p> <p>Generally, if $X$ is a smooth subscheme in $Y$ with normal bundle $N$, then $N_{\tilde{X}}\tilde{Y}=\mathcal{O}_N(-1)$.</p> <p>As for the second one, this is obvious after you combine the comment above with functoriality of the blow-up.</p> http://mathoverflow.net/questions/118584/representations-of-gln-containing-skv/118618#118618 Answer by Anton Fonarev for Representations of $GL(n)$ containing $S^kV$ Anton Fonarev 2013-01-11T11:24:42Z 2013-05-07T22:12:33Z <p>Regarding the question of pairs, here is the answer (hope, I'm not mistaken).</p> <p>Let me first change the notation a little bit. Suppose, we are looking for irreps $V_1=\Sigma^{\lambda}V$ and $V_2=\Sigma^{\mu}V$ where $\lambda = (\lambda_1,\ldots,\lambda_n)$ and $\mu = (\mu_1,\ldots,\mu_n)$ resp., such that there exists an embedding $S^kV\to V_1^*\otimes V_2$.</p> <p>It's immediate that such an embedding exists iff $V_2=\Sigma^{\mu}V$ is an irreducible factor in $V_1\otimes S^kV=\Sigma^{\lambda}V\otimes S^kV$. The Littlewood-Richardson rule gives us the answer. Firstly, $k=|\mu|-|\lambda|$, where $|\lambda| = \sum_i\lambda_i$. Secondly, one has the following inequalities:</p> <ul> <li>$\mu_1\geq \lambda_1$,</li> <li>$\lambda_1\geq \mu_2 \geq \lambda_2$,</li> <li>$\ldots$</li> <li>$\lambda_{n-1}\geq \mu_n \geq \lambda_n$.</li> </ul> <p>Finally, if such an embedding exists, it's unique by the very same L-R rule.</p> <p>UPD: Forgot to mention that <strike>the main question still seems to be very hard as even</strike> checking that $\dim V_1=\dim V_2$ is a huge problem.</p> <p>UPD 2: Let me answer your main question (despite UPD). Let's forget about representation theory and do some linear algebra. What we have is a surjective map $V_1\otimes U\to V_2$, where $\dim V_1 = \dim V_2$. Let the corresponding bilinear map be $h:V_1\times U\to V_2$. We want to find such $u\in U$ that $h_u = h(-, u)$ is invertible, equivalently, of maximal rank.</p> <p>Well, this seems to be quite obvious: let $r$ be the maximal rank of $h_u$ among all $u\in U$. Suppose $r&lt;\dim V_2$. Let $u_0\in U$ be such that $\mathrm{rk}\ h_{u_0}=r$. Take any $v\in V_2\setminus\mathrm{Im}\ h_{u_0}$. Then there exists such $u_1\in U$ that $v\in \mathrm{Im}\ h_{u_1}$. Now it's easy to see that the rank of a general linear combination $\alpha h_{u_0} + \beta h_{u_1} = h_{\alpha u_0+\beta u_1}$ will be greater than $r$.</p> http://mathoverflow.net/questions/109280/completion-of-a-completion/109283#109283 Answer by Anton Fonarev for Completion of a completion Anton Fonarev 2012-10-10T09:27:53Z 2012-10-10T09:27:53Z <p>Let $A$ be a commutative ring, $I\subset A$ be an ideal. Denote by $\hat{A}$ the completion with respect to the $I$-adic filtration.</p> <p>One can consider the kernel $\hat{I}_n$ of the natural projection $\hat{A}\to A/I^n$. The obvious observation is: $\hat{A}/\hat{I}_n=A/I^n$. Thus, $\hat{A}=\varprojlim_n\hat{A}/\hat{I}_n$.</p> <p>Now, it' clear that $I^n\hat{A}\subseteq\hat{I}_n\subseteq(\hat{I}_1)^n$. In Noetherian case they are always equal. This gives an affirmative answer to your question. In general they can differ. However, I can not come up with an example immediately.</p> http://mathoverflow.net/questions/101474/system-of-local-coefficients-on-x-locally-constant-sheaves-and-orientation-sheav/101479#101479 Answer by Anton Fonarev for system of local coefficients on X, locally constant sheaves and orientation sheaves Anton Fonarev 2012-07-06T10:57:29Z 2012-07-06T10:57:29Z <p>These are purely topological notions and have nothing to do with algebraic geometry in particular.</p> <p>Let $M$ for simplicity be a topological manifold of dimension $n$. Then the orientation sheaf $\mathcal{L}_{or}(M)$ is the sheafification of the presheaf $U\mapsto H_n(M,M-U;\mathbb{Z})$. It's always a locally constant sheaf with stalks equal to $\mathbb{Z}$. One immediately checks that $\mathcal{L}_{or}$ is trivial if and only if $M$ is orientable. This definition can be generalized.</p> <p>As for the references, I'd suggest checking A.Dimca, <em>Sheaves in Topology</em> or B.Iversen, <em>Cohomology of Sheaves</em>.</p> http://mathoverflow.net/questions/101294/extension-of-projective-module/101300#101300 Answer by Anton Fonarev for Extension of projective module Anton Fonarev 2012-07-04T10:10:28Z 2012-07-04T10:58:46Z <p>Given a short exact sequence $0\to F_1 \to F \to F_2\to 0$, one has $pd(F)\leq \max \left( pd(F_1),pd(F_2) \right)$ with equality except when $pd(F_2)=pd(F_1)+1$. Suppose that $pd_B(M)&lt;\infty$. Then $pd_B(N) = pd_B(M)$.</p> <p>Now, $N$ is projective if and only if $pd_B(N)=0$. Therefore, one has to ask for $pd_B(M)=0$, which is the same as to say that $M$ is projective as a $B$-module.</p> http://mathoverflow.net/questions/90293/good-overview-of-singularity-theory/90360#90360 Answer by Anton Fonarev for Good overview of singularity theory Anton Fonarev 2012-03-06T13:07:32Z 2012-03-06T13:07:32Z <p>If you are looking for a more topological treatment of the subject, there is a two-volume <strong>Singularities of Differentiable Maps</strong> by Arnold, Varchenko and Gusein-Zade.</p> http://mathoverflow.net/questions/78957/betti-numbers-from-virtual-betti-numbers-of-a-cell-decomposition/78976#78976 Answer by Anton Fonarev for Betti numbers from virtual Betti numbers of a cell decomposition Anton Fonarev 2011-10-24T13:02:35Z 2011-10-25T07:28:02Z <p>Sure, it is true. Let's check once again the definition of virtual Betti numbers:</p> <ol> <li>For $X$ smooth and compact $b_i(X)=\dim_{\mathbb{Q}} H^i(X,\mathbb{Q})$.</li> <li>For any closed $Y\subset X$ one has $b_i(X) = b_i(Y)+b_i(X-Y)$.</li> </ol> <p>Now, whenever you have an algebraic cell decomposition $F^1\subset\ldots\subset F^{n-1}\subset F^n=X$, by (2) you have $b_i(X)=b_i(F^n-F^{n-1})+\ldots+b_i(F^2-F^1)+b_i(F^1)$, where each $F^i-F^{i-1}$ is a disjoint union of some affine spaces. It is left to see that $b_i(\mathbb{A}^j)=b_i(\mathbb{P}^j)-b_i(\mathbb{P}^{j-1})=\delta^i_{2j}$.</p> <p>Now combine with the first Proposition in [Fresse, 4.1] and get that whenever $X$ is projective $b_i(X)=\dim_{\mathbb{Q}}H^i(X,\mathbb{Q})$. This holds because $X$ compact implies $H^i_c(X,\mathbb{Q})=H^i(X,\mathbb{Q})$.</p> <p>Ok, let's see why this Proposition holds in our particular situation. Take the first step of the stratification: $F^{n-1}\subset F^n=X$. Denote $U=F^n-F^{n-1}$. Then there is a long exact sequence associated to a closed subset: $\ldots\to H^i_c(U)\to H^i_c(X)\to H^i_c(F^{n-1})\to H^{i+1}_c(U)\to\ldots$. All the coefficients are $\mathbb{Q}$. Now $U$ is a disjoint union of affine cells of dimension equal to $\dim X$ and $\dim F^{n-1}&lt;\dim X$ (you can always achieve this by a proper choice of stratification). Note that $F^{n-1}$ also admits a cell decomposition. So we may assume that the statement holds for $F^{n-1}$ by an inductive argument. In Particular, $H^i_c(F^{n-1})=0$ for all $i>2\dim F^{n-1}$, thus, for all $i > 2(\dim X-1)$. Combine with the long exact sequence and the fact that $\dim H^i_c(\mathbb{A}^n)=\delta^i_{2n}$. This finishes the proof.</p> <p>The reference for the long exact sequence associated to a closed subset will be Iversen, Cohomology of sheaves.</p> <p>However, as far as I understand, having an algebraic cell decomposition is somewhat strong. Note that in general virtual Betti numbers can be negative.</p> http://mathoverflow.net/questions/77363/action-of-sl-n1-on-couples-of-linear-spaces-in-mathbbpn/77368#77368 Answer by Anton Fonarev for Action of $SL_{n+1}$ on couples of linear spaces in $\mathbb{P}^n$. Anton Fonarev 2011-10-06T15:53:34Z 2011-10-06T15:53:34Z <p>Sure it does. Denote your variety by $X$. Consider this action on the corresponding vector space $V$. Then your projective subspaces correspond to linear subspaces $V_1$ and $V_2$ of dimension $m+1$ and $n-m+1$ that intersect transversely. There is an obvious projection from the variety $Y$ of all frames in $V$ to $X$ (take the linear hull of the first $m+1$ and the last $n-m+1$ vectors). Now, up to a multiplication of all the vectors in the frame by a common scalar, $SL_{n+1}$ acts transitively on $Y$. Thus, it acts transitively on $X$.</p> http://mathoverflow.net/questions/75317/flatness-in-complex-analytic-geometry/75319#75319 Answer by Anton Fonarev for flatness in complex analytic geometry Anton Fonarev 2011-09-13T15:13:52Z 2011-09-13T15:13:52Z <p>Well, I definitely know nothing on the subject. However, there is something called Hironaka division, which gives some way to check flatness explicitly. Check Section 4.2 of these <a href="http://www.math.uwo.ca/~jadamus/papers/AGII.pdf" rel="nofollow">notes</a>.</p> http://mathoverflow.net/questions/74885/exact-sequence-of-logarithmic-differential-sheaves-associated-to-an-effective-car/74947#74947 Answer by Anton Fonarev for exact sequence of logarithmic differential sheaves associated to an effective Cartier divisor on a smooth variety Anton Fonarev 2011-09-08T23:08:05Z 2011-09-09T09:52:34Z <p>Here is a way to define this sheaf algebraically over any field of characteristic zero. Let $\mathrm{T}_{X}$ denote the tangent sheaf on $X$. Choose a local equation $\phi_U$ for $D$ on $U$. Consider the following submodule:</p> <p>$\mathrm{T}_X(-\log\phi_U)=\${$\partial\in\mathrm{Der}(\mathcal{O}_X(U))\mid \partial\phi_U\in (\phi_U)$}$\ \subset \mathrm{T}_X(U)$.</p> <p>It is an easy exercise to check that this does not depend on the choice of equation and glues into a sheaf $\mathrm{T}_X(-\log D)$. Now take the dual $\Omega^1_X(\log D)=\mathrm{T}_X(-\log D)^*$.</p> <p>The good thing about having normal crossings is that in this case $\Omega^1_X(\log D)$ becomes a locally free sheaf.</p> <p>For your second question, look at <strong>Definition 2.5</strong> in this paper by Dolgachev, arXiv:math/0508044</p> http://mathoverflow.net/questions/46541/how-to-introduce-notions-of-flat-projective-and-free-modules/46560#46560 Answer by Anton Fonarev for How to introduce notions of flat, projective and free modules? Anton Fonarev 2010-11-18T23:56:10Z 2010-11-18T23:56:10Z <p>Another student answer:</p> <p>Considering 2b, it seems a bit strange that you want to mention derived functors and not to go into $\mathbf{Tor}$ and $\mathbf{Ext}$. The problem with derived functors is that first of all, categories usually have enough injective objects. Secondly, projectives are just a suitable collection of objects to use, but not always. However, the universal property is really important here. Lifting morphisms is something extremely natural to ask (which <em>is</em> the exactness of $\mathbf{Hom}$). This is also a way to mention that projective modules are direct summands of free (which fits perfectly into you sequence of presentation).</p> <p>Speaking of flat modules, I'd avoid saying about "varying nicely". It's not very convincing the first time you see it (like telling a child that to have a baby one should kiss his partner). I'd also be nice to somehow sneak in the adjointness of $\otimes$ and $\mathbf{Hom}$.</p> http://mathoverflow.net/questions/132965/normal-bundle-of-an-exceptional-divisor/133046#133046 Comment by Anton Fonarev Anton Fonarev 2013-06-07T21:41:23Z 2013-06-07T21:41:23Z By functoriality I mean that the blow up of $X$ is the strict transform of $X$ in the blow up of the ambient $\mathbb{A}^4$. See S&#225;ndor's answer for details. http://mathoverflow.net/questions/118584/representations-of-gln-containing-skv/118618#118618 Comment by Anton Fonarev Anton Fonarev 2013-01-11T13:34:37Z 2013-01-11T13:34:37Z @Klim Due to dualization, in my notation $V_1=S^{k/2}V^*$. In particular, $\lambda = (0,\ldots,0,-k/2)$. http://mathoverflow.net/questions/118584/representations-of-gln-containing-skv Comment by Anton Fonarev Anton Fonarev 2013-01-11T10:59:40Z 2013-01-11T10:59:40Z @Dragos &quot;invariable&quot; should be a misprint for invertible. http://mathoverflow.net/questions/118584/representations-of-gln-containing-skv/118599#118599 Comment by Anton Fonarev Anton Fonarev 2013-01-11T10:57:29Z 2013-01-11T10:57:29Z @Aakumadula It's clearly written that Efim asks for $T(x)$ to be invertible. You can regard $T(x)$ as an element in $Hom(V_1^*,V_2)$. http://mathoverflow.net/questions/114595/can-we-promote-to-a-lie-group-isomorphism Comment by Anton Fonarev Anton Fonarev 2012-11-26T22:41:25Z 2012-11-26T22:41:25Z This has already been around MO: <a href="http://mathoverflow.net/questions/44060" rel="nofollow">mathoverflow.net/questions/44060</a> http://mathoverflow.net/questions/114023/product-of-projective-schemes Comment by Anton Fonarev Anton Fonarev 2012-11-21T10:23:28Z 2012-11-21T10:23:28Z I'm sorry, but this is on the edge of being a research level question (as required in <a href="http://mathoverflow.net/faq" rel="nofollow">mathoverflow.net/faq</a>). http://mathoverflow.net/questions/101474/system-of-local-coefficients-on-x-locally-constant-sheaves-and-orientation-sheav/101479#101479 Comment by Anton Fonarev Anton Fonarev 2012-08-02T12:04:03Z 2012-08-02T12:04:03Z @zatilokum It means that this is a constant sheaf. http://mathoverflow.net/questions/101474/system-of-local-coefficients-on-x-locally-constant-sheaves-and-orientation-sheav Comment by Anton Fonarev Anton Fonarev 2012-07-06T11:00:38Z 2012-07-06T11:00:38Z Could someone retag, please? Say, some &quot;sheaf-theory&quot; and &quot;at.algebraic-topology&quot;. http://mathoverflow.net/questions/101294/extension-of-projective-module/101300#101300 Comment by Anton Fonarev Anton Fonarev 2012-07-04T11:05:49Z 2012-07-04T11:05:49Z @Fernando, this obviously happens as $pd_{\mathbf{Z}/4}(\mathbf{Z}/2)=\infty$. I've already put this assumption that I had in mind. http://mathoverflow.net/questions/101294/extension-of-projective-module/101300#101300 Comment by Anton Fonarev Anton Fonarev 2012-07-04T10:57:14Z 2012-07-04T10:57:14Z @a-fortiori Sure, thanks for pointing this out. http://mathoverflow.net/questions/101294/extension-of-projective-module Comment by Anton Fonarev Anton Fonarev 2012-07-04T10:17:16Z 2012-07-04T10:17:16Z It's somehow weird to have a question starting with &quot;two noncommutative rings&quot; tagged ac.commutative-algebra. Could someone change it to homological? http://mathoverflow.net/questions/100383/do-recoil-rebound-left-mutations-exist-on-a-del-pezzo-surface Comment by Anton Fonarev Anton Fonarev 2012-06-22T23:19:17Z 2012-06-22T23:19:17Z Your definition of an exceptional pair is very special. A usual one asks only for $Ext^\bullet(F, E)=0$. A pair is called strong if $Ext^i(E, F)=0$ for all $i&gt;0$. Also when one defines mutations, $Hom$ is taken in the derived sense. http://mathoverflow.net/questions/94072/dimension-of-fiber-over-projection-of-affine-variety Comment by Anton Fonarev Anton Fonarev 2012-04-14T23:51:58Z 2012-04-14T23:51:58Z This is not an appropriate question for MO. I'm getting more and more concerned with the number of such questions arising in here. http://mathoverflow.net/questions/93389/direct-limits-and-quasi-isomorphism/93402#93402 Comment by Anton Fonarev Anton Fonarev 2012-04-08T15:31:58Z 2012-04-08T15:31:58Z I really like this derived category point of view! http://mathoverflow.net/questions/92354/noether-normalization/92434#92434 Comment by Anton Fonarev Anton Fonarev 2012-03-28T21:58:18Z 2012-03-28T21:58:18Z By the way, if I remember correctly, this is the proof written in Ried's book.